Concept Of Convexity Research Articles

A note on M-convex functions on jump systems

A jump system is defined as a set of integer points (vectors) with a certain exchange property, generalizing the concepts of matroids, delta-matroids, and base polyhedra of integral polymatroids (or submodular systems). A discrete convexity concept is defined for functions on constant-parity jump systems and it has been used in graph theory and algebra. In this paper we call it “jump M-convexity” and extend it to “jump M♮-convexity” for functions defined on a larger class of jump systems. By definition, every jump M-convex function is a jump M♮-convex function, and we show the equivalence of these concepts by establishing an (injective) embedding of jump M♮-convex functions in n variables into the set of jump M-convex functions in n+1 variables. Using this equivalence we show further that jump M♮-convex functions admit a number of natural operations such as aggregation, projection (partial minimization), convolution, composition, and transformation by a network.

Optimal production and inventory control of multi-class mixed backorder and lost sales demand class models

Manufacturers usually face customers with different service requirements. The challenge they face is how to determine which customers to serve when the inventory supply is limited. We consider the problem of a manufacturer serving two types of customers: those with long term commitment, divided into several backorder classes; and those without commitment consisting of a single lost sales class. We treat the problem within a continuous time integrated production and inventory control framework. We propose a two-step strategy to fully characterize the optimal policy. In a first step, we use the concept of L-natural convexity to partially describe the optimal policy. Based on these results, in a second step, we reformulate the problem and fully characterize the optimal policy. We show that the latter is characterized by state-dependent multidimensional thresholds. Due to the computational complexity of the problem, we propose three heuristic policies: The first uses linear thresholds that mimic the optimal policy. These thresholds are computed through a decomposition of the original problem into a series of single-lost sales, single-backorder class problems. The second heuristic treats all demand classes equally. The third heuristic uses static thresholds to control production and inventory rationing among demand classes. Extensive numerical results show that, for problems with one lost sales and two or three backorder classes, the first heuristic outperforms the other two with a cost deviation less than 1.25%, from the optimal. Furthermore, computing the thresholds of this heuristic is orders of magnitude faster than computing the optimal policy.

Strongly Reciprocally p-Convex Functions and Some Inequalities

In this paper, we generalize the concept of strong and reciprocal convexity. Some basic properties and results will be presented for the new class of strongly reciprocally p-convex functions. Furthermore, we will discuss the Hermite–Hadamard-type, Jensen-type, and Fejér-type inequalities for the strongly reciprocally p-convex functions.

The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds

ABSTRACT In this paper, we study the Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function on Hadamard manifolds. The gH-directional differentiability for interval-valued function is defined by using the generalized Hukuhara difference. The concepts of interval-valued convexity and pseudoconvexity are introduced on Hadamard manifolds, and several properties involving such functions are also given. Under these settings, we derive the KKT optimality conditions and give a numerical example to show that the results obtained in this paper are more general than the corresponding conclusions of Wu [The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res. 2007;176:46–59] in solving the optimization problem with interval-valued objective function.

Fuzzy convergence structures in the framework of L-convex spaces

In this paper, fuzzy convergence theory in the framework of $L$-convex spaces is introduced. Firstly, the concept of $L$-convex remote-neighborhood spaces is introduced and it is shown that the resulting category is isomorphic to that of $L$-convex spaces. Secondly, by means of $L$-convex ideals, the notion of $L$-convergence spaces is introduced and it is proved that the category of $L$-convex spaces can be embedded in that of $L$-convergence spaces as a reflective subcategory. Finally, the concepts of convex and preconvex $L$-convergence spaces are introduced and it is shown that the resulting categories are isomorphic to the categories of $L$-convex spaces and $L$-preconvex remote-neighborhood spaces, respectively.

Convexity of sets in metric Abelian groups

Abstract In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

Trapezium-Type Inequalities for Raina’s Fractional Integrals Operator Using Generalized Convex Functions

The authors have reviewed a wide production of scientific articles dealing with the evolution of the concept of convexity and its various applications, and based on this they have detected the relationship that can be established between trapezoidal inequalities, generalized convex functions, and special functions, in particular with the so-called Raina function, which generalizes other better known ones such as the hypergeometric function and the Mittag–Leffler function. The authors approach this situation by studying the Hermite–Hadamard inequality, establishing a useful identity using Raina’s fractional integral operator in the setting of ϕ -convex functions, obtaining some integral inequalities connected with the right-hand side of Hermite–Hadamard-type inequalities for Raina’s fractional integrals. Various special cases have been identified.

Some geometric properties of the non-Newtonian sequence spaces lp (N)

Abstract In this paper, we generalize the concepts of convexity, strict convexity and uniform convexity in the sense of non-Newtonian calculus. The main aim of this study is to obtain the non-Newtonian convexity, non-Newtonian strict convexity and non-Newtonian uniform convexity properties of the non-Newtonian sequence spaces lp (N).

A New Version of the Hermite–Hadamard Inequality for Riemann–Liouville Fractional Integrals

Integral inequalities play a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods. Thus, the present days need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition. There is a strong relationship between convexity and symmetry. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the past few years. In this article, we firstly point out the known Hermite–Hadamard (HH) type inequalities which are related to our main findings. In view of these, we obtain a new inequality of Hermite–Hadamard type for Riemann–Liouville fractional integrals. In addition, we establish a few inequalities of Hermite–Hadamard type for the Riemann integrals and Riemann–Liouville fractional integrals. Finally, three examples are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell discretization. The representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under CFL-like conditions. In contrast to predictor-corrector approaches based on the flux-corrected transport methodology, the proposed flux limiting strategy is monolithic, i.e., limited antidiffusive terms are incorporated into the well-defined residual of a nonlinear (semi-)discrete problem. A stabilized high-order Galerkin discretization is recovered if no limiting is performed. In the limited version, the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors. A formal proof of sparsity is provided for simplicial and box elements. The involved element contributions can be calculated efficiently making use of matrix-free algorithms and precomputed element matrices of the reference element. Numerical studies for Q2 discretizations of linear and nonlinear two-dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions.

Some new Hermite-Hadamard and Fejer type inequalities without convexity/concavity

In this paper we discuss the Hermite-Hadamard and Fejer inequalities vis-a-vis the convexity concept. In particular, we derive some new theorems and examples where Hermite-Hadamard and Fejer type i ...

Hand Gesture Recognition using Convexity Defect

Gestures are the simplest way of conveying a message, rather simpler than verbal means. It is the most primitive way of conversation. Gestures can also be the easiest and intuitive way of communicating with a computer, they can be used to communicate or convey information to computers, robots, smart appliances and many other pieces of machinery. It can eliminate the use of mouse and keyboard to some extent. The gestures cited are basically the variable positions as well as orientations of the hand. They can be detected by a simple webcam attached to the computer. The image is first changed into its corresponding RGB values and then to HSV values for better handling and feature recognition. The hand is segregated from the background using feature extraction. Then the values are matched in proximity of the coded values. Then the region of interest is calculated using the concept of convexity and background subtraction. The convex defect helps to define the contour efficiently. This method is invariant for different positions or direction of the gesture. It is able to detect the number of fingers individually and efficiently.

Editorial

The mathematical theory of Optimal Transport (OT) is now one of the most visible and also most active fields in nonlinear analysis, with two Fieldsmedal winners Cedric Villani and Alessio Figalli among its protagonists. The origins of OT date back to the works of GaspardMonge in the 18th century and those of Leonid Kantorovich in the ’1940s. However, what is considered as the foundations of the modern theory have mostly been established in several ground-breaking works at the end of the 20th century: among the most prominent contributors, there is Yann Brenier, who characterized optimal transport in terms of connections with PDEs and hydrodynamics in the 1980’s; there is RobertMcCann, who introduced the powerful concept of displacement convexity in the 1990’s; there is Felix Otto, who demonstrated the implications of OT to nonlinear evolution equations around the millennium.

CERTAIN INTEGRAL INEQUALITIES CONSIDERING GENERALIZED m-CONVEXITY ON FRACTAL SETS AND THEIR APPLICATIONS

First, we introduce a generalized [Formula: see text]-convexity concept defined on the real linear fractal set [Formula: see text] [Formula: see text] and discuss the relation between generalized [Formula: see text]-convexity and [Formula: see text]-convexity. Second, we present several important properties of the generalized [Formula: see text]-convex mappings. Meanwhile, via local fractional integrals, we also derive certain estimation-type results on generalizations of Hadamard-type, Fejér-type and Simpson-type inequalities. As applications related to local fractional integrals, we construct several inequalities for generalized probability density mappings and [Formula: see text]-type special means.

An excel-based method to enhance the study of “Duration and Convexity” in financial management education

Many concepts in the study of financial management involve complex equations that require manipulating and testing with varying parameters in order to understand the relationships inherent in these equations. Such is the case with the concepts of Bond Duration and Convexity that are taught in such financial management courses as “Financial Futures, Options and Swaps” and “Analysis of Fixed Income Securities” among others. This paper, in comparison to time-consuming mathematical hand calculations, includes Excel-based methods for solving both Duration and Convexity problems in just minutes. In addition, these models are also helpful in classroom settings to explain the effect of interest rate changes on bond price.

Existence and Uniqueness of Zeros for Vector-Valued Functions with K-Adjustability Convexity and Their Applications

In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued functions with K-adjustability convexity. As their applications, we obtain existence theorems for the minimization problem and fixed point problem which are original and quite different from the known results in the existing literature.

A new look at Popoviciu's concept of convexity for functions of two variables

One proves that most results known for the usual convex functions (including Jensen's inequality, Jensen's characterization of convexity, the duality between monotone and convex functions, the existence of affine supports etc.), have analogs for Popoviciu's convex functions. The details are presented in the case of two variables, but the theory exposed here can be easily extended to higher dimensions.

The Convex (\delta,L) Weak Contraction Mapping Theorem and its Non-Self Counterpart in Graphic Language

Let $(X,d)$ be a metric space. A map $T:X \mapsto X$ is said to be a $(\delta,L)$ weak contraction [1] if there exists $\delta \in (0,1)$ and $L\geq 0$ such that the following inequality holds for all $x,y \in X$: $d(Tx,Ty)\leq \delta d (x,y)+Ld(y,Tx)$ On the other hand, the idea of convex contractions appeared in [2] and [3]. In the first part of this paper, motivated by [1]-[3], we introduce a concept of convex $(\delta,L)$ weak contraction, and obtain a fixed point theorem associated with this mapping. In the second part of this paper, we consider the map is a non-self map, and obtain a best proximity point theorem. Finally, we leave the reader with some open problems.

A New Order on Embedded Coalitions: Properties and Applications

Given a finite set of agents, an embedded coalition consists of a coalition and a partition of the rest of agents. We study a partial order on the set of embedded coalitions of a finite set of agents. An embedded coalition precedes another one if the first coalition is contained in the second and the second partition equals the first one after removing the agents in the second coalition. This poset is not a lattice. We describe the maximal lower bounds and minimal upper bounds of a finite subset, whenever they exist. It is a graded poset and we are able to count the number of elements at a given level as well as the total number of chains. The study of this structure allows us to derive results for games with externalities. In particular, we introduce a new concept of convexity and show that it is equivalent to having non-decreasing contributions to embedded coalitions of increasing size.

Convex preferences: A new definition

We suggest a concept of convexity of preferences that does not rely on any algebraic structure. A decision maker has in mind a set of orderings interpreted as evaluation criteria. A preference relation is defined to be convex when it satisfies the following condition: If, for each criterion, there is an element that is both inferior to b by the criterion and superior to a by the preference relation, then b is preferred to a. This definition generalizes the standard Euclidean definition of convex preferences. It is shown that under general conditions, any strict convex preference relation is represented by a maxmin of utility representations of the criteria. Some economic examples are provided.